Second-order Consensus Protocols in Multiple

Vehicle Systems with Local Interactions

Wei Ren∗

Department of Electrical and Computer Engineering, Utah State University, Logan, UT, 84322, USA

Ella Atkins†

Space Systems Laboratory, University of Maryland, College Park, MD 20742, USA

In this paper, a distributed coordination scheme with local interactions is studied formultiple vehicle systems. We introduce a second-order consensus protocol and derivenecessary and/or sufficient conditions under which consensus can be reached in the contextof uni-directional interaction topologies. The consensus protocol is then applied to achievealtitude alignment among a team of micro air vehicles as an illustrative example.

Nomenclature

h Altitude, mλ∗ Autopilot parametersκ∗ Autopilot parametersSubscripti Variable numberSuperscriptc Command

I. Introduction

Cooperative control for multiple vehicle systems has been a topic of significant interest in recent years.For cooperative control strategies to be successful, numerous issues must be addressed, among which

the study of shared information in a group of vehicles facilitates the coordination of these vehicles. As aresult, a critical problem for cooperative control is to design appropriate protocols and algorithms so thatthe group of vehicles can converge to a consistent view of the shared information in the presence of limitedand unreliable information exchange and dynamically changing interaction topologies.

Convergence to a common value is called the consensus or agreement problem in the literature. Consensusproblems have a history in computer science1 and have recently been studied in the context of cooperativecontrol of multiple vehicle systems.2–8 Ref. 9 provides a survey of consensus problems in multi-agent coor-dination. Consensus protocols have potential applications in formation control problems for mobile robots,satellites, or spacecraft and cooperative timing or search missions for multiple unmanned air vehicles. Forexample, information consensus for dynamically evolving information was applied in Ref. 10 to formationflying of multiple space-based interferometers.

One approach to consensus relies on algebraic graph theory, in which graph topologies are connected withthe algebraic properties of the corresponding graph matrices. In Ref. 2 information exchange techniques arestudied to improve stability margins and formation accuracy of vehicle formations. In Ref. 3, sufficientconditions are given for consensus of the heading angles of a group of agents under undirected switching

∗Assistant Professor, Department of Electrical and Computer Engineering, Utah State University, Email: weiren@ieee.org†Assistant Professor, Department of Aerospace Engineering, University of Maryland, Email: ella@ssl.umd.edu, Senior Mem-

ber.

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Copyright © 2005 by Wei Ren and Ella Atkins. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

interaction topologies. In Ref. 4, average consensus problems are solved for a network of integrators us-ing directed graphs. Using directed graphs, Refs. 7 and 8 show necessary and/or sufficient conditions forconsensus of information under time-invariant and switching interaction topologies respectively.

Meanwhile, some other researchers make use of nonlinear mathematical tools to study consensus problems.In Ref. 5, a set-valued Lyapunov approach is used to consider consensus problems with uni-directional time-dependent communication links. In Ref. 11, nonlinear contraction theory is used to study synchronizationand schooling applications, which are related to the consensus problems.

Optimality issues related to consensus problems are also studied in the literature. For example, in Ref. 12,the fastest distributed linear averaging (FDLA) problem are addressed in the context of consensus-seekingamong multiple autonomous agents.

All the previously mentioned references except Ref. 10 focus on consensus protocols that take the formof first-order dynamics. In reality, equations of motion of a broad class of vehicles require second-orderdynamic models. For example, some vehicle dynamics can be feedback linearized as double integrators, e.g.mobile robot dynamic models. In the case of first-order consensus protocols, the final consensus value isa constant. In contrast to the constant final consensus value, it might be proper to derive second-orderconsensus protocols such that some information states converge to a consistent value (e.g. position of theformation center) while others converge to another consistent value (e.g. velocity of the formation center).However, the extension of consensus protocols from first order to second order is nontrivial. In Refs. 10,13–16,formation keeping algorithms taking the form of second-order dynamics are addressed to guarantee attitudealignment, agreement of position deviations and velocities, and/or collision avoidance in a group of vehicles.However, each algorithm mentioned above assumes an undirected interaction topology. The case of directedinteraction topologies is much more challenging than that of undirected interaction topologies. In this paper,we assume a directed interaction topology to take into account the general case where information flow maybe uni-directional. The main contributions of this paper are to introduce a second-order consensus protocoland derive necessary and/or sufficient conditions under which consensus can be reached in the context ofuni-directional interaction topologies.

II. Background and Preliminaries

It is natural to model interaction between vehicles by directed/undirected graphs. A digraph (directedgraph) consists of a pair (N , E), where N is a finite nonempty set of nodes and E ∈ N 2 is a set of orderedpairs of nodes, called edges. As a comparison, the pairs of nodes in an undirected graph are unordered. Ifthere is a directed edge from node vi to node vj , then vi is defined as the parent node and vj is definedas the child node. A directed path is a sequence of ordered edges of the form (vi1 , vi2), (vi2 , vi3), · · · , wherevij ∈ N , in a digraph. An undirected path in an undirected graph is defined accordingly. A digraph is calledstrongly connected if there is a directed path from every node to every other nodes. An undirected graph iscalled connected if there is a path between any distinct pair of nodes. A directed tree is a digraph, whereevery node, except the root, has exactly one parent. A spanning tree of a digraph is a directed tree formedby graph edges that connect all the nodes of the graph. We say that a graph has (or contains) a spanningtree if there exists a spanning tree being a subset of the graph. Note that the condition that a digraph has aspanning tree is equivalent to the case that there exists a node having a directed path to all the other nodes.

GFED@ABCA1++

²²

GFED@ABCA2

²²

kk // GFED@ABCA3

GFED@ABCA4// GFED@ABCA5

// GFED@ABCA6

Figure 1. A digraph that has more than one possible spanning trees, but is not strongly connected.

The adjacency matrix A = [aij ] of a weighted digraph is defined as aii = 0 and aij > 0 if (j, i) ∈ Ewhere i 6= j. The Laplacian matrix of the weighted digraph is defined as L = [`ij ], where `ii =

∑j 6=i aij and

`ij = −aij where i 6= j. For an undirected graph, the Laplacian matrix is symmetric positive semi-definite.

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As an example of a Laplacian matrix for a weighted digraph, the following matrix

L =

1.5 −1.5 0 0 0 0−0.7 0.7 0 0 0 0

0 −1.1 1.1 0 0 0−0.8 0 0 0.8 0 0

0 −0.2 0 −0.3 0.5 00 0 0 0 −1.2 1.2

can be a valid Laplacian matrix corresponding to the digraph in Fig. 1.Let I = {1, 2, · · · , n}. Let 1 and 0 denote the n× 1 column vector of all ones and all zeros respectively.

Let In denote the n×n identity matrix and 0m×n denote the m×n matrix with all zero entries. Let Mn(IR)represent the set of all n × n real matrices. Given a matrix A = [aij ] ∈ Mn(IR), the digraph of A, denotedby Γ(A), is the digraph on n nodes vi, i ∈ I, such that there is a directed edge in Γ(A) from vj to vi if andonly if aij 6= 0 (c.f. Ref. 17).

III. Consensus Protocols

A first-order consensus protocol is proposed in Refs. 3, 4, 6, 7 as

ξi = −n∑

j=1

gijkij(ξi − ξj), i ∈ I (1)

where ξi ∈ IR, kij > 0, gii4= 0, and gij = 1 if information flows from vehicle j to vehicle i and 0 otherwise,

∀i 6= j. The adjacency matrix A of the interaction topology is defined accordingly as aii = 0 and aij = gijkij ,∀i 6= j.

Eq. (1) can be written in matrix form asξ = −Lξ,

where ξ = [ξ1, · · · , ξn]T , and L = [`ij ] is the Laplacian matrix with `ii =∑

j 6=i gijkij and `ij = −gijkij ,∀i 6= j.

The final consensus value using Eq. (1) is given by ξ∗ =∑n

i=1 αiξi(0), where αi ≥ 0 and∑n

i=1 αi = 1.7

Taking into account second-order vehicle dynamics, we propose the following second-order consensusprotocol:

ξi = ζi

ζi = −n∑

j=1

gij [kij(ξi − ξj) + γkij(ζi − ζj)], i ∈ I (2)

where ξi ∈ IR, ζi ∈ IR, kij > 0, γ > 0, gii4= 0, and gij = 1 if information flows from vehicle j to vehicle i

and 0 otherwise, ∀i 6= j.Note that consensus protocols (1) and (2) are distributed in the sense that each vehicle only needs infor-

mation from its (possibly time-varying) local neighbors. The goal of consensus protocol (2) is to guaranteethat |ξi − ξj | → 0 and |ζi − ζj | → 0 as t →∞. In the case that ξi and ζi denote the position and velocity ofthe ith vehicle respectively, Eq. (2) represents the motion of that vehicle.

Let ξ = [ξ1, · · · , ξn]T and ζ = [ζ1, · · · , ζn]T . Eq. (2) can be written in matrix form as[

ξ

ζ

]= Γ

[ξ

ζ

], (3)

where

Γ =

[0n×n In

−L −γL

].

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IV. Convergence Analysis for the Second-order Consensus Protocol

In this paper, we focus on the convergence analysis for consensus protocol (2) under a time-invariantinteraction topology. The convergence analysis for consensus protocol (2) under time-varying interactiontopologies will be addressed in future work.

Given a block matrix

M =

[A B

C D

],

it is known that det(M) = det(AD − CB) if A and C commute, where det(·) denotes the determinant of amatrix.

To find the eigenvalues of Γ, we can solve the equation det(λI2n − Γ) = 0, where det(λI2n − Γ) is thecharacteristic polynomial of matrix Γ. Note that

det(λI2n − Γ) = det

([λIn −In

L λIn + γL

])

= det(λ2In + (1 + γλ)L). (4)

Also note that

det(λIn + L) =n∏

i=1

(λ− µi), (5)

where µi is the ith eigenvalue of −L.By comparing Eqs. (4) and (5), we see that

det(λ2In + (1 + γλ)L

)=

n∏

i=1

(λ2 − (1 + γλ)µi),

which implies that the roots of Eq. (4) can be obtained by solving λ2 = (1 + γλ)µi. Therefore, it isstraightforward to see that the eigenvalues of Γ are given by

λi+ =γµi +

√γ2µ2

i + 4µi

2

λi− =γµi −

√γ2µ2

i + 4µi

2, (6)

where λi+ and λi− are called eigenvalues of Γ that are associated with µi.From Eq. (6), we can see that Γ has 2m zero eigenvalues if and only if −L has m zero eigenvalues. It

is straightforward to see that −L has at least one zero eigenvalue since all its row sums are equal to zero.Therefore, we know that Γ has at least two zero eigenvalues. Without loss of generality, we let λ1+ = λ1− = 0.In addition, we know that all non-zero eigenvalues of −L have negative real parts from the Gersgorin disctheorem.17

We have the following lemma regarding a necessary and sufficient condition for information consensususing consensus protocol (2).

Lemma IV.1 Consensus protocol (2) achieves consensus asymptotically if and only if matrix Γ has exactlytwo zero eigenvalues and all the other eigenvalues have negative real parts. Specifically, ξ → 1pT ξ(0) +t1pT ζ(0) and ζ → 1pT ζ(0), where p is a nonnegative left eigenvector of −L associated with eigenvalue 0 andpT 1 = 1.

Proof: (Sufficiency.) Noting that Γ has two exactly zero eigenvalues, we can verify that eigenvalue zero hasgeometric multiplicity equal to one. As a result, we know that Γ can be written in Jordan canonical form as

Γ = PJP−1

= [w1, · · · , w2n]

0 1 01×(2n−2)

0 0 01×(2n−2)

0(2n−2)×1 0(2n−2)×1 J ′

νT1...

νT2n

, (7)

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where wj , j = 1, · · · , 2n, can be chosen to be the right eigenvectors or generalized eigenvectors of Γ, νTj ,

j = 1, · · · , 2n, can be chosen to be the left eigenvectors or generalized eigenvectors of Γ, and J ′ is the Jordanupper diagonal block matrix corresponding to non-zero eigenvalues λi+ and λi−, i = 2, · · · , n.

Without loss of generality, we choose w1 = [1T ,0T ]T and w2 = [0T ,1T ]T , where it can be verified thatw1 and w2 are an eigenvector and generalized eigenvector of Γ associated with eigenvalue 0 respectively.Noting that Γ has exactly two zero eigenvalues, we know that −L has a simple zero eigenvalue, which inturn implies that there exists a nonnegative vector p such that pT L = 0 and pT 1 = 1 as shown in Ref. 7. Itcan be verified that ν1 = [pT ,0T ]T and ν2 = [0T , pT ]T are a generalized left eigenvector and left eigenvectorof Γ associated with eigenvalue 0 respectively, where νT

1 w1 = 1 and νT2 w2 = 1. Noting that eigenvalues λi+

and λi−, i = 2, · · · , n, have negative real parts, we see that

limt→∞

eΓt = limt→∞

PeJtP−1

= P limt→∞

1 t 01×(2n−2)

0 1 01×(2n−2)

0(2n−2)×1 0(2n−2)×1 eJ ′t

P−1

=

[1pT t1pT

0n×n 1pT

],

where we have used the fact that limt→∞ eJ ′t → 0(2n−2)×(2n−2).Noting that as t →∞ [

ξ(t)ζ(t)

]→

[1pT t1pT

0n×n 1pT

][ξ(0)ζ(0)

],

we see that ξ(t) → 1pT ξ(0) + t1pT ζ(0) and ζ(t) → 1pT ζ(0) as t → ∞. As a result, we know that|ξi(t)− ξj(t)| → 0 and |ζi(t)− ζj(t)| → 0 as t → ∞. That is, consensus is achieved for the group ofvehicles.

(Necessity.) Suppose that the sufficient condition that Γ has exactly two zero eigenvalues and all theother eigenvalues have negative real parts does not hold. Noting that Γ has at least two zero eigenvalues,the fact that the sufficient condition does not hold implies that Γ has either more than two zero eigenvaluesor it has two zero eigenvalues but has at least another eigenvalue having positive real part. In either case, itcan be verified that limt→∞ eJt has a rank larger than two, which implies that limt→∞ eΓt has a rank larger

than two. Note that consensus is reached asymptotically if and only if limt→∞ eΓt →[1pT

1qT

], where p and q

are n× 1 vectors. As a result, the rank of limt→∞ eΓt cannot exceed two. This results in a contradiction.If all non-zero eigenvalues of −L are real and therefore negative, it is straightforward to verify that all

non-zero eigenvalues of Γ have negative real parts following Eq. (6). In the general case, some non-zeroeigenvalues of Γ may have positive real parts even if all non-zero eigenvalues of −L have negative real partsas shown in the following examples.

We consider several cases as follows.Case 1: Interaction Topology Having Separated Subgroups

In the case that the interaction topology has separated subgroups as shown in Fig. 2, consensus cannotbe achieved for the team of vehicles since the information states from different separated groups do not affectone another. In fact, we also know that −L has at least two zero eigenvalues in this case,7 which in turnimplies that Γ has at least four zero eigenvalues.

?>=<89:;A1

²²

?>=<89:;A4

²²?>=<89:;A2?>=<89:;A3

Figure 2. A digraph that has separated subgroups.

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0 1 2 3 4 5 6 7 8 9 10−1

0

1

2

3

4

t (s)

ξ i

ξ1

ξ2

ξ3

ξ4

0 1 2 3 4 5 6 7 8 9 10−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

t (s)

ζ i

ζ1

ζ2

ζ3

ζ4

Figure 3. Evolution of the information states under the interaction topology given by Fig. 2.

Hereafter we assume that kij = 1 and γ = 1 in Eq. (2) unless explicitly mentioned. In addition, we letξi(0) = 0.2(i− 1) and ζi(0) = 0.1(i− 1), i = 1, · · · , 4. Fig. 3 shows the evolution of the information states ξi

and ζi, i = 1, · · · , 4, using consensus protocol (2) under the interaction topology given by Fig. 2. Note thatA1 and A2 reach consensus, and A3 and A4 also reach consensus although the whole group cannot reachconsensus.Case 2: Interaction Topology Having Multiple Leaders

In the case that the interaction topology has multiple leaders as shown in Fig. 4, where both A1 andA4 are leaders, consensus cannot be achieved for the team of vehicles since each leader’s information stateis not affected by any other vehicle’s information state in the team. Noting that −L has at least two rowswith all zero entries in this case, we know that −L has at least two zero eigenvalues, which in turn impliesthat Γ has at least four zero eigenvalues.

?>=<89:;A1

²²

?>=<89:;A4

²²?>=<89:;A2// ?>=<89:;A3

Figure 4. A digraph that has multiple leaders.

Fig. 5 shows the evolution of the information states ξi and ζi, i = 1, · · · , 4, using the consensus protocol (2)under the interaction topology given by Fig. 4. Note that only A1 and A2 reach consensus.Case 3: Connected Undirected Interaction Topology

If the interaction topology is undirected as shown in Fig. 6, we know that the graph Laplacian L issymmetric positive semi-definite, which implies that all eigenvalues of L are real. Therefore, all non-zeroeigenvalues of Γ have negative real parts.

In the case of undirected graphs, graph Laplacian L has a simple zero eigenvalue if and only if the graphis connected. Therefore, we know that consensus is achieved asymptotically if and only if the undirectedgraph is connected.

Fig. 7 shows the evolution of the information states ξi and ζi, i = 1, · · · , 4, using the consensus protocol (2)under the interaction topology given by Fig. 6.Case 4: leader-follower Interaction Topology

In the case that the interaction topology is a leader-follower one as shown in Fig. 8, it is straightforwardto see that L can be written as an upper diagonal matrix by permutation transformations. As a result, weknow that zero is a simple eigenvalue of L and all non-zero eigenvalues are real. Therefore, we know thatconsensus is achieved asymptotically in the case of leader-follower interaction topologies.

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0 1 2 3 4 5 6 7 8 9 10−1

0

1

2

3

4

t (s)

ξ i

ξ1

ξ2

ξ3

ξ4

0 1 2 3 4 5 6 7 8 9 10−0.2

−0.1

0

0.1

0.2

0.3

t (s)

ζ i

ζ1

ζ2

ζ3

ζ4

Figure 5. Evolution of the information states under the interaction topology given by Fig. 4.

?>=<89:;A1?>=<89:;A4

?>=<89:;A2?>=<89:;A3

Figure 6. A connected undirected graph.

Fig. 9 shows the evolution of the information states ξi and ζi, i = 1, · · · , 4, using the consensus protocol (2)under the interaction topology given by Fig. 8.Case 5: Interaction Topology Having a Spanning Tree

Note that the connected undirected topology and the leader following topology can be thought of asspecial cases of an interaction topology having a spanning tree.

In the case that the interaction topology has a spanning tree as shown in Fig. 10, consensus may not beachieved as in the case where the consensus protocol is given by Eq. (1). However, having a spanning treeis a necessary condition for information consensus as will be shown below.

Fig. 11 and 12 show the evolution of the information states ξi and ζi, i = 1, · · · , 4, using the consensusprotocol (2) under the interaction topology given by Fig. 10 with γ = 1 and γ = 0.4 respectively. Note thatconsensus cannot be reached in the case that γ = 0.4. Unlike the previous cases where convergence of theconsensus protocol does not depend upon γ, consensus may not be reached in the general case where theinteraction topology has a spanning tree other than Cases 3 and 4 if γ is too small.

By comparing Figs. 8 and 10, we see that more interactions are involved in Fig. 10 than in Fig. 8 in thesense that A3 sends information to A1 in Fig. 10. However, while the consensus protocol converges underthe interaction topology given by Fig. 8 for any γ > 0, the consensus protocol cannot converge under theinteraction topology given by Fig. 10 if γ is too small. This is somewhat contradictory to our intuition inthe sense that more interactions may lead to instability for the whole group.

In the special case that all eigenvalues of L are real, we have the following lemma.

Lemma IV.2 If −L has a simple zero eigenvalue and all the other eigenvalues are real, consensus proto-col (2) achieves consensus for any γ > 0.

To show that having a spanning tree is a necessary condition for information consensus, we need thefollowing lemma.

Lemma IV.3 7 The graph Laplacian of a directed weighted graph has a simple zero eigenvalue if and onlyif the graph has a spanning tree.

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0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

t (s)

ξ i

ξ1

ξ2

ξ3

ξ4

0 1 2 3 4 5 6 7 8 9 10−0.1

0

0.1

0.2

0.3

0.4

t (s)

ζ i

ζ1

ζ2

ζ3

ζ4

Figure 7. Evolution of the information states under the interaction topology given by Fig. 6.

?>=<89:;A1

²²

// ?>=<89:;A4

?>=<89:;A2// ?>=<89:;A3

Figure 8. A digraph that has a leader following topology.

We have the following necessary condition for information consensus.

Theorem IV.1 Consensus protocol (2) achieves consensus asymptotically only if the interaction topologyhas a spanning tree. a

Proof: If consensus protocol (2) achieves consensus asymptotically, we know that Γ has exactly two zeroeigenvalues following Lemma IV.1. Therefore, we see that matrix L has a simple zero eigenvalue, which inturn implies that the interaction topology has a spanning tree following Lemma IV.3.

Next, we show a sufficient condition for information consensus.

Theorem IV.2 Consensus protocol (2) achieves consensus asymptotically if the interaction topology has aspanning tree and

γ > maxi=2,··· ,n

√2

|µi| cos(π2 − tan−1 −Re(µi)

Im(µi)), (8)

where µi, i = 2, · · · , n, are the non-zero eigenvalues of −L, and Re(·) and Im(·) represent the real andimaginary parts of a number respectively.

Proof: If the interaction topology has a spanning tree, we know that −L has one zero eigenvalue and all theother eigenvalues have negative real parts. Therefore, we know that Γ has two zero eigenvalues. It is left toshow that non-zero eigenvalues of Γ have negative real parts. If inequality (8) is true, we know that λi+ andλi−, i = 2, · · · , n, have negative real parts following the proof of Theorem 6 in Ref. 18, where λi+ and λi−are eigenvalues of Γ associated with µi. As a result, we see that consensus can be achieved asymptoticallyfrom Lemma IV.1.

We also have the following lemma regarding the final consensus value.aAs a comparison, the first-order consensus protocol (1) achieves consensus asymptotically if and only if the interaction

topology has a spanning tree (see Ref. 7).

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0 1 2 3 4 5 6 7 8 9 10−0.2

0

0.2

0.4

0.6

0.8

t (s)

ξ i

ξ1

ξ2

ξ3

ξ4

0 1 2 3 4 5 6 7 8 9 10−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

t (s)

ζ i

ζ1

ζ2

ζ3

ζ4

Figure 9. Evolution of the information states under the interaction topology given by Fig. 8.

?>=<89:;A1

²²

// ?>=<89:;A4

?>=<89:;A2// ?>=<89:;A3

`BBBBBBBBB

Figure 10. A digraph that has a spanning tree.

Lemma IV.4 Suppose that Γ has two zero eigenvalues and all the other eigenvalues have negative real parts.If ζi(0) = 0, i ∈ I, then as t → ∞, ξi(t) →

∑ni=1 piξi(0) and ζi(t) → 0, where i ∈ I and p = [p1, · · · , pn]T

is a nonnegative left eigenvector of −L associated with eigenvalue 0 satisfying∑n

i=1 pi = 1. In addition, ifζi(0) = 0, i ∈ IL, where IL denotes the set of vehicles that have a directed path to all the other vehicles inthe interaction topology, then ξi(t) →

∑i∈IL

piξi(0) and ζi(t) → 0, i ∈ I, as t →∞.

Proof: The first part of the lemma follows directly from the fact that ξ(t) → 1pT ξ(0) + t1pT ζ(0) andζ(t) → 1pT ζ(0) as t →∞.

For the second part of the lemma, we note that pi > 0 if vehicle i has a directed path to all theother vehicles in the interaction topology and pi = 0 otherwise.7 As a result, we know that ξi(t) →∑

i∈ILpiξi(0) + t

∑i∈IL

piζi(0) and ζi(t) →∑

i∈ILpiζi(0) and the second part of the lemma is proved.

Note that ξ → 1pT ξ(0) + t1pT ζ(0) and ζ → 1pT ζ(0) with consensus protocol (2). Under some cir-cumstances, it might be desirable that ξ → 1qT and ζ → 0, where q is an n × 1 vector. For example, information stabilization applications, we want each vehicle to agree on their a priori unknown fixed formationcenter, which has a constant position and zero velocity. In this case, we propose the following second-orderconsensus protocol:

ξi = ζi

ζi = −αζi −n∑

j=1

gij [kij(ξi − ξj) + γkij(ζi − ζj)], (9)

where α > 0.The analysis for consensus protocol (9) is similar to that for consensus protocol (2) and is omitted for

simplicity.

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0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

t (s)

ξ i

ξ1

ξ2

ξ3

ξ4

0 1 2 3 4 5 6 7 8 9 10−0.2

−0.1

0

0.1

0.2

0.3

t (s)

ζ i

ζ1

ζ2

ζ3

ζ4

Figure 11. Evolution of the information states under the interaction topology given by Fig. 10 with γ = 1.

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

t (s)

ξ i

ξ1

ξ2

ξ3

ξ4

0 1 2 3 4 5 6 7 8 9 10−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

t (s)

ζ i

ζ1

ζ2

ζ3

ζ4

Figure 12. Evolution of the information states under the interaction topology given by Fig. 10 with γ = 0.4.

V. Illustrative Example

In this section, we apply the second-order consensus protocol to achieve altitude alignment among multiplemicro unmanned air vehicles.

Let hi denote the altitude of the ith unmanned air vehicle (UAV). For UAVs equipped with efficient low-level autopilots, the resulting UAV/autopilot models are assumed to be second order for altitude hold.19,20

For a fixed-wing micro air vehicle, the simplified equation of motion for altitude is given by

hi = −λhihi + λhi(hci − hi), (10)

where hci is the altitude command to the low-level controllers, and λ∗ are positive constants.19

For a rotary-wing micro air vehicle, the simplified equation of motion for altitude is given by

hi = κhi(hci − hi) (11)

where hci is the vertical velocity command to the low-level controllers, and κhi is a positive constant.20

Let νi be defined as

νi = −αhi −n∑

j=1

gijkij(hi − hj)−n∑

j=1

gijγkij(hi − hj),

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where α ≥ 0, kij > 0, γ > 0, and gij is defined the same as in Eq. (2).For Eq. (10), we propose the altitude command as

hci = hi +

λhi

λhihi +

1λhi

νi. (12)

For Eq. (11), we propose the vertical velocity command as

hci = hi +

1κhi

νi. (13)

In the following, we only consider altitude alignment for multiple rotary-wing micro air vehicles withcontrol law (13). Results for multiple fixed-wing micro air vehicles with control law (12) are similar. Theinteraction topology for the micro air vehicles is given by Fig. 13, where a directed edge from the jth vehicle tothe ith vehicle means that the ith vehicle can obtain hj and hj from the jth vehicle through a uni-directionalcommunication link. Note that Fig. 13 has a spanning tree.

76540123V1// 76540123V2

²²

// 76540123V3

76540123V4

OO

76540123V5//oo 76540123V6

Figure 13. The interaction topology between the six micro air vehicles.

We assume that κhi = 1, and the vertical velocity command is saturated and satisfies |hci | ≤ 0.5 m/s.

In the first case, we let α = 0, kij = 1, and γ = 1, which guarantees that Γ has two zero eigenvalues andall the other eigenvalues have negative real parts. Fig. 14 shows the altitudes and vertical velocities of eachvehicle. Note that altitude is aligned between those vehicles. Fig. 15 shows the vertical velocity commandsof each vehicle.

0 5 10 15 2098

100

102

104

t (s)

altit

ude

(m)

# 1# 2# 3# 4# 5# 6

0 5 10 15 20−0.5

0

0.5

t (s)

vert

ical

vel

ocity

(m

/s) # 1

# 2# 3# 4# 5# 6

Figure 14. Altitudes and vertical velocities of each vehicle with γ = 1.

As a comparison, we let α = 0, kij = 1, and γ = 0.1 in the second case. It can be shown that twoeigenvalues of Γ have positive real parts, which implies that consensus cannot be achieved. Fig. 16 shows thealtitudes and vertical velocities of each vehicle. Note that altitude cannot be aligned between those vehiclesin this case. Fig. 17 shows the vertical velocity commands of each vehicle.

VI. Conclusion

We have proposed a second-order protocol for information consensus among multiple vehicles. We havealso shown necessary and/or sufficient conditions under which consensus can be achieved in the context ofuni-directional interaction topologies. The second-order consensus protocol has been applied to align thealtitudes of multiple rotary-wing micro air vehicles in a distributed manner as a proof of concept.

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0 5 10 15 20−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

t (s)co

mm

ande

d ve

rtic

al v

eloc

ity (

m/s

)

# 1# 2# 3# 4# 5# 6

Figure 15. Commanded vertical velocities of each vehicle with γ = 1.

0 5 10 15 2098

100

102

104

t (s)

altit

ude

(m)

# 1# 2# 3# 4# 5# 6

0 5 10 15 20−0.5

0

0.5

t (s)

vert

ical

vel

ocity

(m

/s) # 1

# 2# 3# 4# 5# 6

Figure 16. Altitudes and vertical velocities of each vehicle with γ = 0.1.

Acknowledgments

This research is supported by the Army Research Office through the MAV MURI Program (Grant No.ARMY-W911NF0410176) with Technical Monitor as Dr. Gary Anderson.

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0 5 10 15 20−0.5

−0.4

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